Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $q = \dfrac{k^2 - 4k - 12}{2k^2 - 34k + 140} \div \dfrac{2k^2 + 4k}{k^2 - 10k} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{k^2 - 4k - 12}{2k^2 - 34k + 140} \times \dfrac{k^2 - 10k}{2k^2 + 4k} $ First factor out any common factors. $q = \dfrac{k^2 - 4k - 12}{2(k^2 - 17k + 70)} \times \dfrac{k(k - 10)}{2k(k + 2)} $ Then factor the quadratic expressions. $q = \dfrac {(k + 2)(k - 6)} {2(k - 10)(k - 7)} \times \dfrac {k(k - 10)} {2k(k + 2)} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac { (k + 2)(k - 6) \times k(k - 10)} { 2(k - 10)(k - 7) \times 2k(k + 2)} $ $q = \dfrac {k(k + 2)(k - 6)(k - 10)} {4k(k - 10)(k - 7)(k + 2)} $ Notice that $(k - 10)$ and $(k + 2)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac {k(k + 2)(k - 6)\cancel{(k - 10)}} {4k\cancel{(k - 10)}(k - 7)(k + 2)} $ We are dividing by $k - 10$ , so $k - 10 \neq 0$ Therefore, $k \neq 10$ $q = \dfrac {k\cancel{(k + 2)}(k - 6)\cancel{(k - 10)}} {4k\cancel{(k - 10)}(k - 7)\cancel{(k + 2)}} $ We are dividing by $k + 2$ , so $k + 2 \neq 0$ Therefore, $k \neq -2$ $q = \dfrac {k(k - 6)} {4k(k - 7)} $ $ q = \dfrac{k - 6}{4(k - 7)}; k \neq 10; k \neq -2 $